About direct sum/product and subdirect product of rings

Question

I'm studying ring theory and I'm stuck in the topic of direct product of rings. If someone can help me, here are my questions: Is there something called direct sum in the category of unital rings?, if the answer is no, what is the coproduct of (unital) rings? Also, is valid the notion of subdirect product in rings with unity? Finally, is there a book where I can study these topics?

I only have two books of ring theory where the authors study direct sums/products of rings, but they don't assume that rings have $1$, so I'm having trouble understanding the definitions. Thanks in advance.

Answer 1

Is there something called direct sum in the category of unital rings?

The earliest use of the phrase "direct sum" that can be located (today) on MathSciNet is in the paper:

Subrings of Direct Sums by Neal H. McCoy, American Journal of Mathematics, Vol. 60, No. 2 (Apr., 1938), pp. 374-382.

The definition given for a direct sum of rings is: "If $K$ is any ring, a direct sum of rings $K$ is understood to be the ring of all functions with values in $K$, defined on some finite or infinite set $M$." Under this definition, $R$ is a direct sum of copies of $K$ if it is a subring of $K^M$ for some $M$.

Within a short time "direct sum" was being used as a synonym for "direct product" in the case of finitely many factors, e.g. in

Dyer-Bennet, John A note on finite regular rings. Bull. Amer. Math. Soc. 47, (1941). 784-787.

Nowadays it seems that ring theorists still use direct sum as a synonym for direct product in the case where there are finitely many factors.

if the answer is no, what is the coproduct of (unital) rings?

For an equationally definable class of algebras, the forgetful functor creates products, which implies that the definition of product in such a category does not change when you enlarge or shrink the category. But the forgetful functor does not create coproducts. The construction of coproducts depends on the context. In the category of commutative unital rings, the coproduct of $R$ and $S$ is $R\otimes_{\mathbb Z} S$. In the category of all unital rings it is something different.

One way to deal with coproducts in equational definable categories of algebras is through presentations. If $A$ is presented by $\langle G\;|\;R\rangle$, $B$ is presented by $\langle H\;|\;S\rangle$ and $G\cap H=\emptyset$, then $A\sqcup B$ has presentation $\langle G\cup H\;|\;R\cup S\rangle$.

Also, is valid the notion of subdirect product in rings with unity?

Yes, a subobject of a product is subdirect if it projects onto any coordinate.

is there a book where I can study these topics?

Bergman, Clifford, Universal algebra. Fundamentals and selected topics. Pure and Applied Mathematics (Boca Raton), 301. CRC Press, Boca Raton, FL, 2012.

Dauns, John, Modules and rings. Cambridge University Press, Cambridge, 1994.

Answer 2

Is there something called direct sum in the category of unital rings?

No, there is no operation on rings that deserves to be called the direct sum.

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if the answer is no, what is the coproduct of (unital) rings?

Here is a way to construct the coproduct of a family of rings $R_i$ in the category $\mathsf{Ring}$:

Let $S = \mathbb{Z} \langle x_{i, r} : i \in I, r \in R_i \rangle$ be the free noncommutative ring on the indeterminates $x_{i, r}$ for $i \in I$ and $r \in R_i$. Now let $J \subseteq S$ be the ideal generated by expressions of the following form:

• $x_{i, r_1} + x_{i, r_2} - x_{i, r_1 + r_2}$ for $i \in I$ and $r_1, r_2 \in R_i$
• $x_{i, r_1} x_{i, r_2} - x_{i, r_1 r_2}$ for $i \in I$ and $r_1, r_2 \in R_i$
• $x_{i, 1} - 1$ for $i \in I$

Now take $\newcommand\quotient[2]{{^{\Large #1}}/{_{ \Large #2}}} T = \quotient{S}{J}$. Then $T \cong \coprod_{i \in I} R_i$ is the coproduct over all the $R_i$ in $\mathsf{Ring}$. The inclusions $R_i \to T$ are the obvious ones, and the fact that we took the quotient by $J$ ensures that these are actually ring homomorphisms.

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Also, is valid the notion of subdirect product in rings with unity?

Unfortunately, I don't know anything about subdirect products.

Finally, is there a book where I can study these topics?

Unfortunately, I have no good suggestions for books on these topics.

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Here are some related questions:

• Infinite coproduct of rings
• Coproduct in the category of (noncommutative) associative algebras
• How to construct the coproduct of two (non-commutative) rings
• How to construct the tensor product of two (non-commutative) algebras?
• Coproduct of non-unital commutative algebras